Can neural networks efficiently solve the traveling salesmen problem?
To the best of my knowledge, there isn't any difference between the algorithmic methods and the NN methods. Those that can solve in polynomial time do not give a precise solution. Those that do give a precise solution do not solve in polynomial time. Of those that give a precise solution, the fastest takes $2^N$, but it blows up in terms of memory. The fastest good algorithm I believe is Concorde.
The efficient algorithms solve in polynomial time, don't blow up in terms of memory, and give a solution close to perfect, say, within 2-3%. Again, to the best of my knowledge, no NN has beaten the best algorithmic solutions, but there are suggestions that some NN solution could be faster.